What's the underlying idea of definition of constrained market in Skiadas' Asset Pricing Theory?

Question

I'm self-studying Skiadas' Asset Pricing Theory, and find the definition of constrained market on page 21 confusing(you can find it here in the sample chapter).

Definition 1.26. A constrained market is a closed convex set of cash flows $X \subseteq \Bbb R^{1+K}$ such that $0 \in X$ and for some $\epsilon > 0$,

$x \in X$ and $0 < \| x \| <\epsilon$ implies $\frac{\epsilon}{\| x \|}x \in X.$

I know this definition renders missing market and short-sale constraints as special cases, but the underlying idea of this formulation still eludes me.

Answer 1

I have asked myself the very same question when I first read the book. As far as I can tell, the "scalability" condition is only imposed for technical reasons. It simplifies the subsequent proof of the Fundemental Theorem of Asset Pricing in constrained markets.

There are several papers that have shown that the theorem is valid for conic constraints. Examples are Napp or Pham & Touzi. The main result from Napp shows what steps are necessary to obtain the theorem with general conic constraints on the amount that investors are able to hold of each asset.